Average Error: 34.4 → 10.4
Time: 5.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.0674124610604968 \cdot 10^{-82}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.96876625840091586 \cdot 10^{107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{b_2} \cdot \sqrt[3]{b_2}, -\sqrt[3]{b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.0674124610604968 \cdot 10^{-82}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 5.96876625840091586 \cdot 10^{107}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{b_2} \cdot \sqrt[3]{b_2}, -\sqrt[3]{b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r16326 = b_2;
        double r16327 = -r16326;
        double r16328 = r16326 * r16326;
        double r16329 = a;
        double r16330 = c;
        double r16331 = r16329 * r16330;
        double r16332 = r16328 - r16331;
        double r16333 = sqrt(r16332);
        double r16334 = r16327 - r16333;
        double r16335 = r16334 / r16329;
        return r16335;
}
↓
double f(double a, double b_2, double c) {
        double r16336 = b_2;
        double r16337 = -1.0674124610604968e-82;
        bool r16338 = r16336 <= r16337;
        double r16339 = -0.5;
        double r16340 = c;
        double r16341 = r16340 / r16336;
        double r16342 = r16339 * r16341;
        double r16343 = 5.968766258400916e+107;
        bool r16344 = r16336 <= r16343;
        double r16345 = cbrt(r16336);
        double r16346 = r16345 * r16345;
        double r16347 = -r16345;
        double r16348 = r16336 * r16336;
        double r16349 = a;
        double r16350 = r16349 * r16340;
        double r16351 = r16348 - r16350;
        double r16352 = sqrt(r16351);
        double r16353 = -r16352;
        double r16354 = fma(r16346, r16347, r16353);
        double r16355 = r16354 / r16349;
        double r16356 = 0.5;
        double r16357 = r16356 * r16341;
        double r16358 = 2.0;
        double r16359 = r16336 / r16349;
        double r16360 = r16358 * r16359;
        double r16361 = r16357 - r16360;
        double r16362 = r16344 ? r16355 : r16361;
        double r16363 = r16338 ? r16342 : r16362;
        return r16363;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -1.0674124610604968e-82

    1. Initial program 52.3

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 8.8

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

    if -1.0674124610604968e-82 < b_2 < 5.968766258400916e+107

    1. Initial program 13.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt13.9

      \[\leadsto \frac{\left(-\color{blue}{\left(\sqrt[3]{b_2} \cdot \sqrt[3]{b_2}\right) \cdot \sqrt[3]{b_2}}\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    4. Applied distribute-rgt-neg-in13.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{b_2} \cdot \sqrt[3]{b_2}\right) \cdot \left(-\sqrt[3]{b_2}\right)} - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    5. Applied fma-neg13.9

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{b_2} \cdot \sqrt[3]{b_2}, -\sqrt[3]{b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a}\]

    if 5.968766258400916e+107 < b_2

    1. Initial program 50.0

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 3.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.0674124610604968 \cdot 10^{-82}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.96876625840091586 \cdot 10^{107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{b_2} \cdot \sqrt[3]{b_2}, -\sqrt[3]{b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))